src/HOL/Library/Topology_Euclidean_Space.thy
author huffman
Thu, 04 Jun 2009 17:24:09 -0700
changeset 31447 97bab1ac463e
parent 31445 c8a474a919a7
child 31448 29090e3111bd
permissions -rw-r--r--
generalize type of 'at' to topological_space; generalize some lemmas
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
     1
(* Title:      Topology
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
     2
   Author:     Amine Chaieb, University of Cambridge
30267
171b3bd93c90 removed old/broken CVS Ids;
wenzelm
parents: 30262
diff changeset
     3
   Author:     Robert Himmelmann, TU Muenchen
171b3bd93c90 removed old/broken CVS Ids;
wenzelm
parents: 30262
diff changeset
     4
*)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
     5
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
     6
header {* Elementary topology in Euclidean space. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
     7
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
     8
theory Topology_Euclidean_Space
30654
254478a8dd05 dropped theory Arith_Tools
haftmann
parents: 30582
diff changeset
     9
imports SEQ Euclidean_Space
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    10
begin
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    11
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    12
declare fstcart_pastecart[simp] sndcart_pastecart[simp]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    13
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    14
subsection{* General notion of a topology *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    15
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    16
definition "istopology L \<longleftrightarrow> {} \<in> L \<and> (\<forall>S \<in>L. \<forall>T \<in>L. S \<inter> T \<in> L) \<and> (\<forall>K. K \<subseteq>L \<longrightarrow> \<Union> K \<in> L)"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
    17
typedef (open) 'a topology = "{L::('a set) set. istopology L}"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    18
  morphisms "openin" "topology"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    19
  unfolding istopology_def by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    20
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    21
lemma istopology_open_in[intro]: "istopology(openin U)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    22
  using openin[of U] by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    23
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    24
lemma topology_inverse': "istopology U \<Longrightarrow> openin (topology U) = U"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    25
  using topology_inverse[unfolded mem_def Collect_def] .
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    26
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    27
lemma topology_inverse_iff: "istopology U \<longleftrightarrow> openin (topology U) = U"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    28
  using topology_inverse[of U] istopology_open_in[of "topology U"] by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    29
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    30
lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    31
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    32
  {assume "T1=T2" hence "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    33
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    34
  {assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    35
    hence "openin T1 = openin T2" by (metis mem_def set_ext)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    36
    hence "topology (openin T1) = topology (openin T2)" by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    37
    hence "T1 = T2" unfolding openin_inverse .}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    38
  ultimately show ?thesis by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    39
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    40
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    41
text{* Infer the "universe" from union of all sets in the topology. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    42
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    43
definition "topspace T =  \<Union>{S. openin T S}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    44
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    45
subsection{* Main properties of open sets *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    46
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    47
lemma openin_clauses:
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    48
  fixes U :: "'a topology"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    49
  shows "openin U {}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    50
  "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    51
  "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    52
  using openin[of U] unfolding istopology_def Collect_def mem_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    53
  by (metis mem_def subset_eq)+
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    54
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    55
lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    56
  unfolding topspace_def by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    57
lemma openin_empty[simp]: "openin U {}" by (simp add: openin_clauses)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    58
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    59
lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    60
  by (simp add: openin_clauses)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    61
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    62
lemma openin_Union[intro]: "(\<forall>S \<in>K. openin U S) \<Longrightarrow> openin U (\<Union> K)" by (simp add: openin_clauses)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    63
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    64
lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    65
  using openin_Union[of "{S,T}" U] by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    66
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    67
lemma openin_topspace[intro, simp]: "openin U (topspace U)" by (simp add: openin_Union topspace_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    68
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    69
lemma openin_subopen: "openin U S \<longleftrightarrow> (\<forall>x \<in> S. \<exists>T. openin U T \<and> x \<in> T \<and> T \<subseteq> S)" (is "?lhs \<longleftrightarrow> ?rhs")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    70
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    71
  {assume ?lhs then have ?rhs by auto }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    72
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    73
  {assume H: ?rhs
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
    74
    then obtain t where t: "\<forall>x\<in>S. openin U (t x) \<and> x \<in> t x \<and> t x \<subseteq> S"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    75
      unfolding Ball_def ex_simps(6)[symmetric] choice_iff by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    76
    from t have th0: "\<forall>x\<in> t`S. openin U x" by auto
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
    77
    have "\<Union> t`S = S" using t by auto
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    78
    with openin_Union[OF th0] have "openin U S" by simp }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    79
  ultimately show ?thesis by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    80
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    81
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    82
subsection{* Closed sets *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    83
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    84
definition "closedin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> openin U (topspace U - S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    85
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    86
lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U" by (metis closedin_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    87
lemma closedin_empty[simp]: "closedin U {}" by (simp add: closedin_def)
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
    88
lemma closedin_topspace[intro,simp]:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    89
  "closedin U (topspace U)" by (simp add: closedin_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    90
lemma closedin_Un[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<union> T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    91
  by (auto simp add: Diff_Un closedin_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    92
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    93
lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union> {A - s|s. s\<in>S}" by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    94
lemma closedin_Inter[intro]: assumes Ke: "K \<noteq> {}" and Kc: "\<forall>S \<in>K. closedin U S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    95
  shows "closedin U (\<Inter> K)"  using Ke Kc unfolding closedin_def Diff_Inter by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    96
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    97
lemma closedin_Int[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<inter> T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    98
  using closedin_Inter[of "{S,T}" U] by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
    99
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   100
lemma Diff_Diff_Int: "A - (A - B) = A \<inter> B" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   101
lemma openin_closedin_eq: "openin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> closedin U (topspace U - S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   102
  apply (auto simp add: closedin_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   103
  apply (metis openin_subset subset_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   104
  apply (auto simp add: Diff_Diff_Int)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   105
  apply (subgoal_tac "topspace U \<inter> S = S")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   106
  by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   107
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   108
lemma openin_closedin:  "S \<subseteq> topspace U \<Longrightarrow> (openin U S \<longleftrightarrow> closedin U (topspace U - S))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   109
  by (simp add: openin_closedin_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   110
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   111
lemma openin_diff[intro]: assumes oS: "openin U S" and cT: "closedin U T" shows "openin U (S - T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   112
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   113
  have "S - T = S \<inter> (topspace U - T)" using openin_subset[of U S]  oS cT
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   114
    by (auto simp add: topspace_def openin_subset)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   115
  then show ?thesis using oS cT by (auto simp add: closedin_def)
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   116
qed
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   117
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   118
lemma closedin_diff[intro]: assumes oS: "closedin U S" and cT: "openin U T" shows "closedin U (S - T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   119
proof-
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   120
  have "S - T = S \<inter> (topspace U - T)" using closedin_subset[of U S]  oS cT
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   121
    by (auto simp add: topspace_def )
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   122
  then show ?thesis using oS cT by (auto simp add: openin_closedin_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   123
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   124
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   125
subsection{* Subspace topology. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   126
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   127
definition "subtopology U V = topology {S \<inter> V |S. openin U S}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   128
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   129
lemma istopology_subtopology: "istopology {S \<inter> V |S. openin U S}" (is "istopology ?L")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   130
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   131
  have "{} \<in> ?L" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   132
  {fix A B assume A: "A \<in> ?L" and B: "B \<in> ?L"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   133
    from A B obtain Sa and Sb where Sa: "openin U Sa" "A = Sa \<inter> V" and Sb: "openin U Sb" "B = Sb \<inter> V" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   134
    have "A\<inter>B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)"  using Sa Sb by blast+
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   135
    then have "A \<inter> B \<in> ?L" by blast}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   136
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   137
  {fix K assume K: "K \<subseteq> ?L"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   138
    have th0: "?L = (\<lambda>S. S \<inter> V) ` openin U "
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   139
      apply (rule set_ext)
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   140
      apply (simp add: Ball_def image_iff)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   141
      by (metis mem_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   142
    from K[unfolded th0 subset_image_iff]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   143
    obtain Sk where Sk: "Sk \<subseteq> openin U" "K = (\<lambda>S. S \<inter> V) ` Sk" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   144
    have "\<Union>K = (\<Union>Sk) \<inter> V" using Sk by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   145
    moreover have "openin U (\<Union> Sk)" using Sk by (auto simp add: subset_eq mem_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   146
    ultimately have "\<Union>K \<in> ?L" by blast}
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   147
  ultimately show ?thesis unfolding istopology_def by blast
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   148
qed
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   149
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   150
lemma openin_subtopology:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   151
  "openin (subtopology U V) S \<longleftrightarrow> (\<exists> T. (openin U T) \<and> (S = T \<inter> V))"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   152
  unfolding subtopology_def topology_inverse'[OF istopology_subtopology]
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   153
  by (auto simp add: Collect_def)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   154
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   155
lemma topspace_subtopology: "topspace(subtopology U V) = topspace U \<inter> V"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   156
  by (auto simp add: topspace_def openin_subtopology)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   157
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   158
lemma closedin_subtopology:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   159
  "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   160
  unfolding closedin_def topspace_subtopology
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   161
  apply (simp add: openin_subtopology)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   162
  apply (rule iffI)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   163
  apply clarify
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   164
  apply (rule_tac x="topspace U - T" in exI)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   165
  by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   166
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   167
lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   168
  unfolding openin_subtopology
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   169
  apply (rule iffI, clarify)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   170
  apply (frule openin_subset[of U])  apply blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   171
  apply (rule exI[where x="topspace U"])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   172
  by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   173
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   174
lemma subtopology_superset: assumes UV: "topspace U \<subseteq> V"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   175
  shows "subtopology U V = U"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   176
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   177
  {fix S
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   178
    {fix T assume T: "openin U T" "S = T \<inter> V"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   179
      from T openin_subset[OF T(1)] UV have eq: "S = T" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   180
      have "openin U S" unfolding eq using T by blast}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   181
    moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   182
    {assume S: "openin U S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   183
      hence "\<exists>T. openin U T \<and> S = T \<inter> V"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   184
	using openin_subset[OF S] UV by auto}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   185
    ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S" by blast}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   186
  then show ?thesis unfolding topology_eq openin_subtopology by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   187
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   188
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   189
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   190
lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   191
  by (simp add: subtopology_superset)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   192
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   193
lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   194
  by (simp add: subtopology_superset)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   195
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   196
subsection{* The universal Euclidean versions are what we use most of the time *}
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   197
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   198
definition
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   199
  "open" :: "'a::topological_space set \<Rightarrow> bool" where
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   200
  "open S \<longleftrightarrow> S \<in> topo"
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   201
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   202
definition
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   203
  closed :: "'a::topological_space set \<Rightarrow> bool" where
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   204
  "closed S \<longleftrightarrow> open(UNIV - S)"
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   205
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   206
definition
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   207
  euclidean :: "'a::topological_space topology" where
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   208
  "euclidean = topology open"
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   209
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   210
lemma open_UNIV[intro,simp]:  "open UNIV"
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   211
  unfolding open_def by (rule topo_UNIV)
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   212
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   213
lemma open_inter[intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<inter> T)"
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   214
  unfolding open_def by (rule topo_Int)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   215
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   216
lemma open_Union[intro]: "(\<forall>S\<in>K. open S) \<Longrightarrow> open (\<Union> K)"
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   217
  unfolding open_def subset_eq [symmetric] by (rule topo_Union)
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   218
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   219
lemma open_empty[intro,simp]: "open {}"
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   220
  using open_Union [of "{}"] by simp
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   221
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   222
lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   223
  unfolding euclidean_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   224
  apply (rule cong[where x=S and y=S])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   225
  apply (rule topology_inverse[symmetric])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   226
  apply (auto simp add: istopology_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   227
  by (auto simp add: mem_def subset_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   228
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   229
lemma topspace_euclidean: "topspace euclidean = UNIV"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   230
  apply (simp add: topspace_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   231
  apply (rule set_ext)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   232
  by (auto simp add: open_openin[symmetric])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   233
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   234
lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   235
  by (simp add: topspace_euclidean topspace_subtopology)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   236
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   237
lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   238
  by (simp add: closed_def closedin_def topspace_euclidean open_openin)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   239
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   240
lemma open_Un[intro]:
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   241
  fixes S T :: "'a::topological_space set"
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   242
  shows "open S \<Longrightarrow> open T \<Longrightarrow> open (S\<union>T)"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   243
  by (auto simp add: open_openin)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   244
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   245
lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   246
  by (simp add: open_openin openin_subopen[symmetric])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   247
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   248
lemma closed_empty[intro, simp]: "closed {}" by (simp add: closed_closedin)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   249
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   250
lemma closed_UNIV[simp,intro]: "closed UNIV"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   251
  by (simp add: closed_closedin topspace_euclidean[symmetric])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   252
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   253
lemma closed_Un[intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S\<union>T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   254
  by (auto simp add: closed_closedin)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   255
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   256
lemma closed_Int[intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S\<inter>T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   257
  by (auto simp add: closed_closedin)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   258
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   259
lemma closed_Inter[intro]: assumes H: "\<forall>S \<in>K. closed S" shows "closed (\<Inter>K)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   260
  using H
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   261
  unfolding closed_closedin
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   262
  apply (cases "K = {}")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   263
  apply (simp add: closed_closedin[symmetric])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   264
  apply (rule closedin_Inter, auto)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   265
  done
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   266
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   267
lemma open_closed: "open S \<longleftrightarrow> closed (UNIV - S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   268
  by (simp add: open_openin closed_closedin topspace_euclidean openin_closedin_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   269
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   270
lemma closed_open: "closed S \<longleftrightarrow> open(UNIV - S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   271
  by (simp add: open_openin closed_closedin topspace_euclidean closedin_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   272
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   273
lemma open_diff[intro]: "open S \<Longrightarrow> closed T \<Longrightarrow> open (S - T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   274
  by (auto simp add: open_openin closed_closedin)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   275
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   276
lemma closed_diff[intro]: "closed S \<Longrightarrow> open T \<Longrightarrow> closed(S-T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   277
  by (auto simp add: open_openin closed_closedin)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   278
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   279
lemma open_Inter[intro]: assumes fS: "finite S" and h: "\<forall>T\<in>S. open T" shows "open (\<Inter>S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   280
  using h by (induct rule: finite_induct[OF fS], auto)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   281
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   282
lemma closed_Union[intro]: assumes fS: "finite S" and h: "\<forall>T\<in>S. closed T" shows "closed (\<Union>S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   283
  using h by (induct rule: finite_induct[OF fS], auto)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   284
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   285
subsection{* Open and closed balls. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   286
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   287
definition
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   288
  ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   289
  "ball x e = {y. dist x y < e}"
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   290
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   291
definition
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   292
  cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   293
  "cball x e = {y. dist x y \<le> e}"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   294
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   295
lemma mem_ball[simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e" by (simp add: ball_def)
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   296
lemma mem_cball[simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e" by (simp add: cball_def)
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   297
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   298
lemma mem_ball_0 [simp]:
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   299
  fixes x :: "'a::real_normed_vector"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   300
  shows "x \<in> ball 0 e \<longleftrightarrow> norm x < e"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   301
  by (simp add: dist_norm)
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   302
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   303
lemma mem_cball_0 [simp]:
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   304
  fixes x :: "'a::real_normed_vector"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   305
  shows "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   306
  by (simp add: dist_norm)
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   307
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   308
lemma centre_in_cball[simp]: "x \<in> cball x e \<longleftrightarrow> 0\<le> e"  by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   309
lemma ball_subset_cball[simp,intro]: "ball x e \<subseteq> cball x e" by (simp add: subset_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   310
lemma subset_ball[intro]: "d <= e ==> ball x d \<subseteq> ball x e" by (simp add: subset_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   311
lemma subset_cball[intro]: "d <= e ==> cball x d \<subseteq> cball x e" by (simp add: subset_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   312
lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   313
  by (simp add: expand_set_eq) arith
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   314
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   315
lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   316
  by (simp add: expand_set_eq)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   317
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   318
subsection{* Topological properties of open balls *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   319
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   320
lemma open_dist:
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   321
  fixes S :: "'a::metric_space set"
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   322
  shows "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>x'. dist x' x < e \<longrightarrow> x' \<in> S)"
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   323
  unfolding open_def topo_dist by simp
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   324
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   325
lemma diff_less_iff: "(a::real) - b > 0 \<longleftrightarrow> a > b"
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   326
  "(a::real) - b < 0 \<longleftrightarrow> a < b"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   327
  "a - b < c \<longleftrightarrow> a < c +b" "a - b > c \<longleftrightarrow> a > c +b" by arith+
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   328
lemma diff_le_iff: "(a::real) - b \<ge> 0 \<longleftrightarrow> a \<ge> b" "(a::real) - b \<le> 0 \<longleftrightarrow> a \<le> b"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   329
  "a - b \<le> c \<longleftrightarrow> a \<le> c +b" "a - b \<ge> c \<longleftrightarrow> a \<ge> c +b"  by arith+
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   330
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   331
lemma open_ball[intro, simp]: "open (ball x e)"
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   332
  unfolding open_dist ball_def Collect_def Ball_def mem_def
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   333
  unfolding dist_commute
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   334
  apply clarify
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   335
  apply (rule_tac x="e - dist xa x" in exI)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   336
  using dist_triangle_alt[where z=x]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   337
  apply (clarsimp simp add: diff_less_iff)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   338
  apply atomize
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   339
  apply (erule_tac x="x'" in allE)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   340
  apply (erule_tac x="xa" in allE)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   341
  by arith
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   342
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   343
lemma centre_in_ball[simp]: "x \<in> ball x e \<longleftrightarrow> e > 0" by (metis mem_ball dist_self)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   344
lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   345
  unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   346
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   347
lemma open_contains_ball_eq: "open S \<Longrightarrow> \<forall>x. x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   348
  by (metis open_contains_ball subset_eq centre_in_ball)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   349
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   350
lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   351
  unfolding mem_ball expand_set_eq
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   352
  apply (simp add: not_less)
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   353
  by (metis zero_le_dist order_trans dist_self)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   354
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   355
lemma ball_empty[intro]: "e \<le> 0 ==> ball x e = {}" by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   356
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   357
subsection{* Basic "localization" results are handy for connectedness. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   358
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   359
lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   360
  by (auto simp add: openin_subtopology open_openin[symmetric])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   361
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   362
lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   363
  by (auto simp add: openin_open)
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   364
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   365
lemma open_openin_trans[trans]:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   366
 "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   367
  by (metis Int_absorb1  openin_open_Int)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   368
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   369
lemma open_subset:  "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   370
  by (auto simp add: openin_open)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   371
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   372
lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   373
  by (simp add: closedin_subtopology closed_closedin Int_ac)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   374
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   375
lemma closedin_closed_Int: "closed S ==> closedin (subtopology euclidean U) (U \<inter> S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   376
  by (metis closedin_closed)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   377
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   378
lemma closed_closedin_trans: "closed S \<Longrightarrow> closed T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> closedin (subtopology euclidean S) T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   379
  apply (subgoal_tac "S \<inter> T = T" )
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   380
  apply auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   381
  apply (frule closedin_closed_Int[of T S])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   382
  by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   383
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   384
lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   385
  by (auto simp add: closedin_closed)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   386
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   387
lemma openin_euclidean_subtopology_iff:
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   388
  fixes S U :: "'a::metric_space set"
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   389
  shows "openin (subtopology euclidean U) S
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   390
  \<longleftrightarrow> S \<subseteq> U \<and> (\<forall>x\<in>S. \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x'\<in> S)" (is "?lhs \<longleftrightarrow> ?rhs")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   391
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   392
  {assume ?lhs hence ?rhs unfolding openin_subtopology open_openin[symmetric]
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   393
      by (simp add: open_dist) blast}
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   394
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   395
  {assume SU: "S \<subseteq> U" and H: "\<And>x. x \<in> S \<Longrightarrow> \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x' \<in> S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   396
    from H obtain d where d: "\<And>x . x\<in> S \<Longrightarrow> d x > 0 \<and> (\<forall>x' \<in> U. dist x' x < d x \<longrightarrow> x' \<in> S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   397
      by metis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   398
    let ?T = "\<Union>{B. \<exists>x\<in>S. B = ball x (d x)}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   399
    have oT: "open ?T" by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   400
    { fix x assume "x\<in>S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   401
      hence "x \<in> \<Union>{B. \<exists>x\<in>S. B = ball x (d x)}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   402
	apply simp apply(rule_tac x="ball x(d x)" in exI) apply auto
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   403
        by (rule d [THEN conjunct1])
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   404
      hence "x\<in> ?T \<inter> U" using SU and `x\<in>S` by auto  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   405
    moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   406
    { fix y assume "y\<in>?T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   407
      then obtain B where "y\<in>B" "B\<in>{B. \<exists>x\<in>S. B = ball x (d x)}" by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   408
      then obtain x where "x\<in>S" and x:"y \<in> ball x (d x)" by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   409
      assume "y\<in>U"
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   410
      hence "y\<in>S" using d[OF `x\<in>S`] and x by(auto simp add: dist_commute) }
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   411
    ultimately have "S = ?T \<inter> U" by blast
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   412
    with oT have ?lhs unfolding openin_subtopology open_openin[symmetric] by blast}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   413
  ultimately show ?thesis by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   414
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   415
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   416
text{* These "transitivity" results are handy too. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   417
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   418
lemma openin_trans[trans]: "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   419
  \<Longrightarrow> openin (subtopology euclidean U) S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   420
  unfolding open_openin openin_open by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   421
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   422
lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   423
  by (auto simp add: openin_open intro: openin_trans)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   424
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   425
lemma closedin_trans[trans]:
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   426
 "closedin (subtopology euclidean T) S \<Longrightarrow>
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   427
           closedin (subtopology euclidean U) T
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   428
           ==> closedin (subtopology euclidean U) S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   429
  by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   430
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   431
lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   432
  by (auto simp add: closedin_closed intro: closedin_trans)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   433
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   434
subsection{* Connectedness *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   435
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   436
definition "connected S \<longleftrightarrow>
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   437
  ~(\<exists>e1 e2. open e1 \<and> open e2 \<and> S \<subseteq> (e1 \<union> e2) \<and> (e1 \<inter> e2 \<inter> S = {})
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   438
  \<and> ~(e1 \<inter> S = {}) \<and> ~(e2 \<inter> S = {}))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   439
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   440
lemma connected_local:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   441
 "connected S \<longleftrightarrow> ~(\<exists>e1 e2.
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   442
                 openin (subtopology euclidean S) e1 \<and>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   443
                 openin (subtopology euclidean S) e2 \<and>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   444
                 S \<subseteq> e1 \<union> e2 \<and>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   445
                 e1 \<inter> e2 = {} \<and>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   446
                 ~(e1 = {}) \<and>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   447
                 ~(e2 = {}))"
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   448
unfolding connected_def openin_open by (safe, blast+)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   449
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   450
lemma exists_diff: "(\<exists>S. P(UNIV - S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   451
proof-
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   452
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   453
  {assume "?lhs" hence ?rhs by blast }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   454
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   455
  {fix S assume H: "P S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   456
    have "S = UNIV - (UNIV - S)" by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   457
    with H have "P (UNIV - (UNIV - S))" by metis }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   458
  ultimately show ?thesis by metis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   459
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   460
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   461
lemma connected_clopen: "connected S \<longleftrightarrow>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   462
        (\<forall>T. openin (subtopology euclidean S) T \<and>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   463
            closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   464
proof-
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   465
  have " \<not> connected S \<longleftrightarrow> (\<exists>e1 e2. open e1 \<and> open (UNIV - e2) \<and> S \<subseteq> e1 \<union> (UNIV - e2) \<and> e1 \<inter> (UNIV - e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (UNIV - e2) \<inter> S \<noteq> {})"
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   466
    unfolding connected_def openin_open closedin_closed
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   467
    apply (subst exists_diff) by blast
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   468
  hence th0: "connected S \<longleftrightarrow> \<not> (\<exists>e2 e1. closed e2 \<and> open e1 \<and> S \<subseteq> e1 \<union> (UNIV - e2) \<and> e1 \<inter> (UNIV - e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (UNIV - e2) \<inter> S \<noteq> {})"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   469
    (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)") apply (simp add: closed_def) by metis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   470
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   471
  have th1: "?rhs \<longleftrightarrow> \<not> (\<exists>t' t. closed t'\<and>t = S\<inter>t' \<and> t\<noteq>{} \<and> t\<noteq>S \<and> (\<exists>t'. open t' \<and> t = S \<inter> t'))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   472
    (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   473
    unfolding connected_def openin_open closedin_closed by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   474
  {fix e2
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   475
    {fix e1 have "?P e2 e1 \<longleftrightarrow> (\<exists>t.  closed e2 \<and> t = S\<inter>e2 \<and> open e1 \<and> t = S\<inter>e1 \<and> t\<noteq>{} \<and> t\<noteq>S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   476
	by auto}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   477
    then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by metis}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   478
  then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   479
  then show ?thesis unfolding th0 th1 by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   480
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   481
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   482
lemma connected_empty[simp, intro]: "connected {}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   483
  by (simp add: connected_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   484
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   485
subsection{* Hausdorff and other separation properties *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   486
31421
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   487
axclass t0_space \<subseteq> topological_space
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   488
  t0_space:
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   489
    "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> \<not> (x \<in> U \<longleftrightarrow> y \<in> U)"
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   490
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   491
axclass t1_space \<subseteq> topological_space
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   492
  t1_space:
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   493
    "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<notin> U \<and> x \<notin> V \<and> y \<in> V"
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   494
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   495
instance t1_space \<subseteq> t0_space
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   496
by default (fast dest: t1_space)
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   497
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   498
text {* T2 spaces are also known as Hausdorff spaces. *}
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   499
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   500
axclass t2_space \<subseteq> topological_space
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   501
  hausdorff:
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   502
    "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   503
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   504
instance t2_space \<subseteq> t1_space
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   505
by default (fast dest: hausdorff)
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   506
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   507
instance metric_space \<subseteq> t2_space
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   508
proof
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   509
  fix x y :: "'a::metric_space"
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   510
  assume xy: "x \<noteq> y"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   511
  let ?U = "ball x (dist x y / 2)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   512
  let ?V = "ball y (dist x y / 2)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   513
  have th0: "\<And>d x y z. (d x z :: real) <= d x y + d y z \<Longrightarrow> d y z = d z y
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   514
               ==> ~(d x y * 2 < d x z \<and> d z y * 2 < d x z)" by arith
31421
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   515
  have "open ?U \<and> open ?V \<and> x \<in> ?U \<and> y \<in> ?V \<and> ?U \<inter> ?V = {}"
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   516
    using dist_pos_lt[OF xy] th0[of dist,OF dist_triangle dist_commute]
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   517
    by (auto simp add: expand_set_eq)
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   518
  then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   519
    by blast
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   520
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   521
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   522
lemma separation_t2:
31421
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   523
  fixes x y :: "'a::t2_space"
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   524
  shows "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {})"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   525
  using hausdorff[of x y] by blast
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   526
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   527
lemma separation_t1:
31421
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   528
  fixes x y :: "'a::t1_space"
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   529
  shows "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in>U \<and> y\<notin> U \<and> x\<notin>V \<and> y\<in>V)"
31421
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   530
  using t1_space[of x y] by blast
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   531
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   532
lemma separation_t0:
31421
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   533
  fixes x y :: "'a::t0_space"
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   534
  shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> ~(x\<in>U \<longleftrightarrow> y\<in>U))"
1501fc26f11b add classes for t0, t1, and t2 spaces
huffman
parents: 31420
diff changeset
   535
  using t0_space[of x y] by blast
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   536
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   537
subsection{* Limit points *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   538
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   539
definition
31420
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   540
  islimpt:: "'a::topological_space \<Rightarrow> 'a set \<Rightarrow> bool"
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   541
    (infixr "islimpt" 60) where
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   542
  "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   543
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   544
  (* FIXME: Sure this form is OK????*)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   545
lemma islimptE: assumes "x islimpt S" and "x \<in> T" and "open T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   546
  obtains "(\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   547
  using assms unfolding islimpt_def by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   548
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   549
lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T ==> x islimpt T" by (auto simp add: islimpt_def)
31420
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   550
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   551
lemma islimpt_approachable:
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   552
  fixes x :: "'a::metric_space"
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   553
  shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   554
  unfolding islimpt_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   555
  apply auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   556
  apply(erule_tac x="ball x e" in allE)
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   557
  apply auto
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   558
  apply(rule_tac x=y in bexI)
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   559
  apply (auto simp add: dist_commute)
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   560
  apply (simp add: open_dist, drule (1) bspec)
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   561
  apply (clarify, drule spec, drule (1) mp, auto)
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   562
  done
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   563
31420
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   564
lemma islimpt_approachable_le:
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   565
  fixes x :: "'a::metric_space"
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   566
  shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x <= e)"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   567
  unfolding islimpt_approachable
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   568
  using approachable_lt_le[where f="\<lambda>x'. dist x' x" and P="\<lambda>x'. \<not> (x'\<in>S \<and> x'\<noteq>x)"]
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   569
  by metis (* FIXME: VERY slow! *)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   570
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   571
axclass perfect_space \<subseteq> metric_space
31420
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   572
  (* FIXME: perfect_space should inherit from topological_space *)
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   573
  islimpt_UNIV [simp, intro]: "x islimpt UNIV"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   574
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   575
lemma perfect_choose_dist:
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   576
  fixes x :: "'a::perfect_space"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   577
  shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   578
using islimpt_UNIV [of x]
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   579
by (simp add: islimpt_approachable)
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   580
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   581
instance real :: perfect_space
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   582
apply default
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   583
apply (rule islimpt_approachable [THEN iffD2])
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   584
apply (clarify, rule_tac x="x + e/2" in bexI)
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   585
apply (auto simp add: dist_norm)
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   586
done
31420
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   587
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   588
instance "^" :: (perfect_space, finite) perfect_space
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   589
proof
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   590
  fix x :: "'a ^ 'b"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   591
  {
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   592
    fix e :: real assume "0 < e"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   593
    def a \<equiv> "x $ arbitrary"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   594
    have "a islimpt UNIV" by (rule islimpt_UNIV)
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   595
    with `0 < e` obtain b where "b \<noteq> a" and "dist b a < e"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   596
      unfolding islimpt_approachable by auto
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   597
    def y \<equiv> "Cart_lambda ((Cart_nth x)(arbitrary := b))"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   598
    from `b \<noteq> a` have "y \<noteq> x"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   599
      unfolding a_def y_def by (simp add: Cart_eq)
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   600
    from `dist b a < e` have "dist y x < e"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   601
      unfolding dist_vector_def a_def y_def
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   602
      apply simp
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   603
      apply (rule le_less_trans [OF setL2_le_setsum [OF zero_le_dist]])
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   604
      apply (subst setsum_diff1' [where a=arbitrary], simp, simp, simp)
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   605
      done
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   606
    from `y \<noteq> x` and `dist y x < e`
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   607
    have "\<exists>y\<in>UNIV. y \<noteq> x \<and> dist y x < e" by auto
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   608
  }
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   609
  then show "x islimpt UNIV" unfolding islimpt_approachable by blast
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   610
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   611
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   612
lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   613
  unfolding closed_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   614
  apply (subst open_subopen)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   615
  apply (simp add: islimpt_def subset_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   616
  by (metis DiffE DiffI UNIV_I insertCI insert_absorb mem_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   617
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   618
lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
31420
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   619
  unfolding islimpt_def by auto
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   620
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   621
lemma closed_positive_orthant: "closed {x::real^'n::finite. \<forall>i. 0 \<le>x$i}"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   622
proof-
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   623
  let ?U = "UNIV :: 'n set"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   624
  let ?O = "{x::real^'n. \<forall>i. x$i\<ge>0}"
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   625
  {fix x:: "real^'n" and i::'n assume H: "\<forall>e>0. \<exists>x'\<in>?O. x' \<noteq> x \<and> dist x' x < e"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   626
    and xi: "x$i < 0"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   627
    from xi have th0: "-x$i > 0" by arith
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   628
    from H[rule_format, OF th0] obtain x' where x': "x' \<in>?O" "x' \<noteq> x" "dist x' x < -x $ i" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   629
      have th:" \<And>b a (x::real). abs x <= b \<Longrightarrow> b <= a ==> ~(a + x < 0)" by arith
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   630
      have th': "\<And>x (y::real). x < 0 \<Longrightarrow> 0 <= y ==> abs x <= abs (y - x)" by arith
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   631
      have th1: "\<bar>x$i\<bar> \<le> \<bar>(x' - x)$i\<bar>" using x'(1) xi
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   632
	apply (simp only: vector_component)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   633
	by (rule th') auto
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
   634
      have th2: "\<bar>dist x x'\<bar> \<ge> \<bar>(x' - x)$i\<bar>" using  component_le_norm[of "x'-x" i]
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31286
diff changeset
   635
	apply (simp add: dist_norm) by norm
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   636
      from th[OF th1 th2] x'(3) have False by (simp add: dist_commute) }
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   637
  then show ?thesis unfolding closed_limpt islimpt_approachable
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   638
    unfolding not_le[symmetric] by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   639
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   640
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31286
diff changeset
   641
lemma finite_set_avoid:
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   642
  fixes a :: "'a::metric_space"
31289
847f00f435d4 move dist operation to new metric_space class
huffman
parents: 31286
diff changeset
   643
  assumes fS: "finite S" shows  "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   644
proof(induct rule: finite_induct[OF fS])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   645
  case 1 thus ?case apply auto by ferrack
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   646
next
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   647
  case (2 x F)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   648
  from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   649
  {assume "x = a" hence ?case using d by auto  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   650
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   651
  {assume xa: "x\<noteq>a"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   652
    let ?d = "min d (dist a x)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   653
    have dp: "?d > 0" using xa d(1) using dist_nz by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   654
    from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x" by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   655
    with dp xa have ?case by(auto intro!: exI[where x="?d"]) }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   656
  ultimately show ?case by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   657
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   658
31420
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   659
lemma islimpt_finite:
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   660
  fixes S :: "'a::metric_space set"
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   661
  assumes fS: "finite S" shows "\<not> a islimpt S"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   662
  unfolding islimpt_approachable
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   663
  using finite_set_avoid[OF fS, of a] by (metis dist_commute  not_le)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   664
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   665
lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   666
  apply (rule iffI)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   667
  defer
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   668
  apply (metis Un_upper1 Un_upper2 islimpt_subset)
31420
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   669
  unfolding islimpt_def
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   670
  apply (rule ccontr, clarsimp, rename_tac A B)
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   671
  apply (drule_tac x="A \<inter> B" in spec)
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   672
  apply (auto simp add: open_inter)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   673
  done
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   674
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   675
lemma discrete_imp_closed:
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   676
  fixes S :: "'a::metric_space set"
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   677
  assumes e: "0 < e" and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   678
  shows "closed S"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   679
proof-
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   680
  {fix x assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   681
    from e have e2: "e/2 > 0" by arith
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   682
    from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y\<noteq>x" "dist y x < e/2" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   683
    let ?m = "min (e/2) (dist x y) "
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   684
    from e2 y(2) have mp: "?m > 0" by (simp add: dist_nz[THEN sym])
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   685
    from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z\<noteq>x" "dist z x < ?m" by blast
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   686
    have th: "dist z y < e" using z y
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   687
      by (intro dist_triangle_lt [where z=x], simp)
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   688
    from d[rule_format, OF y(1) z(1) th] y z
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   689
    have False by (auto simp add: dist_commute)}
31420
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   690
  then show ?thesis by (metis islimpt_approachable closed_limpt [where 'a='a])
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   691
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   692
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   693
subsection{* Interior of a Set *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   694
definition "interior S = {x. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   695
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   696
lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   697
  apply (simp add: expand_set_eq interior_def)
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   698
  apply (subst (2) open_subopen) by (safe, blast+)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   699
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   700
lemma interior_open: "open S ==> (interior S = S)" by (metis interior_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   701
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   702
lemma interior_empty[simp]: "interior {} = {}" by (simp add: interior_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   703
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   704
lemma open_interior[simp, intro]: "open(interior S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   705
  apply (simp add: interior_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   706
  apply (subst open_subopen) by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   707
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   708
lemma interior_interior[simp]: "interior(interior S) = interior S" by (metis interior_eq open_interior)
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   709
lemma interior_subset: "interior S \<subseteq> S" by (auto simp add: interior_def)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   710
lemma subset_interior: "S \<subseteq> T ==> (interior S) \<subseteq> (interior T)" by (auto simp add: interior_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   711
lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T ==> T \<subseteq> (interior S)" by (auto simp add: interior_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   712
lemma interior_unique: "T \<subseteq> S \<Longrightarrow> open T  \<Longrightarrow> (\<forall>T'. T' \<subseteq> S \<and> open T' \<longrightarrow> T' \<subseteq> T) \<Longrightarrow> interior S = T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   713
  by (metis equalityI interior_maximal interior_subset open_interior)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   714
lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e. 0 < e \<and> ball x e \<subseteq> S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   715
  apply (simp add: interior_def)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   716
  by (metis open_contains_ball centre_in_ball open_ball subset_trans)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   717
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   718
lemma open_subset_interior: "open S ==> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   719
  by (metis interior_maximal interior_subset subset_trans)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   720
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   721
lemma interior_inter[simp]: "interior(S \<inter> T) = interior S \<inter> interior T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   722
  apply (rule equalityI, simp)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   723
  apply (metis Int_lower1 Int_lower2 subset_interior)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   724
  by (metis Int_mono interior_subset open_inter open_interior open_subset_interior)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   725
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   726
lemma interior_limit_point [intro]:
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   727
  fixes x :: "'a::perfect_space"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   728
  assumes x: "x \<in> interior S" shows "x islimpt S"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   729
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   730
  from x obtain e where e: "e>0" "\<forall>x'. dist x x' < e \<longrightarrow> x' \<in> S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   731
    unfolding mem_interior subset_eq Ball_def mem_ball by blast
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   732
  {
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   733
    fix d::real assume d: "d>0"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   734
    let ?m = "min d e"
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   735
    have mde2: "0 < ?m" using e(1) d(1) by simp
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   736
    from perfect_choose_dist [OF mde2, of x]
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   737
    obtain y where "y \<noteq> x" and "dist y x < ?m" by blast
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   738
    then have "dist y x < e" "dist y x < d" by simp_all
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   739
    from `dist y x < e` e(2) have "y \<in> S" by (simp add: dist_commute)
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   740
    have "\<exists>x'\<in>S. x'\<noteq> x \<and> dist x' x < d"
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   741
      using `y \<in> S` `y \<noteq> x` `dist y x < d` by fast
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   742
  }
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   743
  then show ?thesis unfolding islimpt_approachable by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   744
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   745
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   746
lemma interior_closed_Un_empty_interior:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   747
  assumes cS: "closed S" and iT: "interior T = {}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   748
  shows "interior(S \<union> T) = interior S"
31394
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   749
proof
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   750
  show "interior S \<subseteq> interior (S\<union>T)"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   751
    by (rule subset_interior, blast)
31394
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   752
next
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   753
  show "interior (S \<union> T) \<subseteq> interior S"
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   754
  proof
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   755
    fix x assume "x \<in> interior (S \<union> T)"
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   756
    then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T"
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   757
      unfolding interior_def by fast
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   758
    show "x \<in> interior S"
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   759
    proof (rule ccontr)
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   760
      assume "x \<notin> interior S"
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   761
      with `x \<in> R` `open R` obtain y where "y \<in> R - S"
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   762
        unfolding interior_def expand_set_eq by fast
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   763
      from `open R` `closed S` have "open (R - S)" by (rule open_diff)
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   764
      from `R \<subseteq> S \<union> T` have "R - S \<subseteq> T" by fast
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   765
      from `y \<in> R - S` `open (R - S)` `R - S \<subseteq> T` `interior T = {}`
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   766
      show "False" unfolding interior_def by fast
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   767
    qed
8d8417abb14f generalize lemma interior_closed_Un_empty_interior
huffman
parents: 31393
diff changeset
   768
  qed
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   769
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   770
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   771
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   772
subsection{* Closure of a Set *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   773
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   774
definition "closure S = S \<union> {x | x. x islimpt S}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   775
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   776
lemma closure_interior: "closure S = UNIV - interior (UNIV - S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   777
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   778
  { fix x
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   779
    have "x\<in>UNIV - interior (UNIV - S) \<longleftrightarrow> x \<in> closure S"  (is "?lhs = ?rhs")
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   780
    proof
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   781
      let ?exT = "\<lambda> y. (\<exists>T. open T \<and> y \<in> T \<and> T \<subseteq> UNIV - S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   782
      assume "?lhs"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   783
      hence *:"\<not> ?exT x"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   784
	unfolding interior_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   785
	by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   786
      { assume "\<not> ?rhs"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   787
	hence False using *
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   788
	  unfolding closure_def islimpt_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   789
	  by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   790
      }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   791
      thus "?rhs"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   792
	by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   793
    next
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   794
      assume "?rhs" thus "?lhs"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   795
	unfolding closure_def interior_def islimpt_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   796
	by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   797
    qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   798
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   799
  thus ?thesis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   800
    by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   801
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   802
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   803
lemma interior_closure: "interior S = UNIV - (closure (UNIV - S))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   804
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   805
  { fix x
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   806
    have "x \<in> interior S \<longleftrightarrow> x \<in> UNIV - (closure (UNIV - S))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   807
      unfolding interior_def closure_def islimpt_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   808
      by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   809
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   810
  thus ?thesis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   811
    by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   812
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   813
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   814
lemma closed_closure[simp, intro]: "closed (closure S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   815
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   816
  have "closed (UNIV - interior (UNIV -S))" by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   817
  thus ?thesis using closure_interior[of S] by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   818
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   819
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   820
lemma closure_hull: "closure S = closed hull S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   821
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   822
  have "S \<subseteq> closure S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   823
    unfolding closure_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   824
    by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   825
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   826
  have "closed (closure S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   827
    using closed_closure[of S]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   828
    by assumption
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   829
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   830
  { fix t
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   831
    assume *:"S \<subseteq> t" "closed t"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   832
    { fix x
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   833
      assume "x islimpt S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   834
      hence "x islimpt t" using *(1)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   835
	using islimpt_subset[of x, of S, of t]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   836
	by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   837
    }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   838
    with * have "closure S \<subseteq> t"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   839
      unfolding closure_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   840
      using closed_limpt[of t]
31345
80667d5bee32 generalize topological notions to class metric_space; add class perfect_space
huffman
parents: 31344
diff changeset
   841
      by auto
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   842
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   843
  ultimately show ?thesis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   844
    using hull_unique[of S, of "closure S", of closed]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   845
    unfolding mem_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   846
    by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   847
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   848
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   849
lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   850
  unfolding closure_hull
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   851
  using hull_eq[of closed, unfolded mem_def, OF  closed_Inter, of S]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   852
  by (metis mem_def subset_eq)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   853
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   854
lemma closure_closed[simp]: "closed S \<Longrightarrow> closure S = S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   855
  using closure_eq[of S]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   856
  by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   857
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   858
lemma closure_closure[simp]: "closure (closure S) = closure S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   859
  unfolding closure_hull
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   860
  using hull_hull[of closed S]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   861
  by assumption
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   862
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   863
lemma closure_subset: "S \<subseteq> closure S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   864
  unfolding closure_hull
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   865
  using hull_subset[of S closed]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   866
  by assumption
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   867
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   868
lemma subset_closure: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   869
  unfolding closure_hull
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   870
  using hull_mono[of S T closed]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   871
  by assumption
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   872
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   873
lemma closure_minimal: "S \<subseteq> T \<Longrightarrow>  closed T \<Longrightarrow> closure S \<subseteq> T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   874
  using hull_minimal[of S T closed]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   875
  unfolding closure_hull mem_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   876
  by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   877
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   878
lemma closure_unique: "S \<subseteq> T \<and> closed T \<and> (\<forall> T'. S \<subseteq> T' \<and> closed T' \<longrightarrow> T \<subseteq> T') \<Longrightarrow> closure S = T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   879
  using hull_unique[of S T closed]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   880
  unfolding closure_hull mem_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   881
  by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   882
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   883
lemma closure_empty[simp]: "closure {} = {}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   884
  using closed_empty closure_closed[of "{}"]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   885
  by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   886
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   887
lemma closure_univ[simp]: "closure UNIV = UNIV"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   888
  using closure_closed[of UNIV]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   889
  by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   890
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   891
lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   892
  using closure_empty closure_subset[of S]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   893
  by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   894
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   895
lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   896
  using closure_eq[of S] closure_subset[of S]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   897
  by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   898
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   899
lemma open_inter_closure_eq_empty:
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   900
  "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   901
  using open_subset_interior[of S "UNIV - T"]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   902
  using interior_subset[of "UNIV - T"]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   903
  unfolding closure_interior
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   904
  by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   905
31420
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   906
lemma open_inter_closure_subset:
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   907
  fixes S :: "'a::metric_space set"
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   908
    (* FIXME: generalize to topological_space *)
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   909
  shows "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   910
proof
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   911
  fix x
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   912
  assume as: "open S" "x \<in> S \<inter> closure T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   913
  { assume *:"x islimpt T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   914
    { fix e::real
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   915
      assume "e > 0"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   916
      from as `open S` obtain e' where "e' > 0" and e':"\<forall>x'. dist x' x < e' \<longrightarrow> x' \<in> S"
31418
9baa48bad81c generalize some constants and lemmas to class topological_space
huffman
parents: 31402
diff changeset
   917
	unfolding open_dist
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   918
	by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   919
      let ?e = "min e e'"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   920
      from `e>0` `e'>0` have "?e > 0"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   921
	by simp
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   922
      then obtain y where y:"y\<in>T" "y \<noteq> x \<and> dist y x < ?e"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   923
	using islimpt_approachable[of x T] using *
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   924
	by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   925
      hence "\<exists>x'\<in>S \<inter> T. x' \<noteq> x \<and> dist x' x < e" using e'
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   926
	using y
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   927
	by(rule_tac x=y in bexI, simp+)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   928
    }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   929
    hence "x islimpt S \<inter> T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   930
      using islimpt_approachable[of x "S \<inter> T"]
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   931
      by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   932
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   933
  then show "x \<in> closure (S \<inter> T)" using as
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   934
    unfolding closure_def
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   935
    by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   936
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   937
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   938
lemma closure_complement: "closure(UNIV - S) = UNIV - interior(S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   939
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   940
  have "S = UNIV - (UNIV - S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   941
    by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   942
  thus ?thesis
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   943
    unfolding closure_interior
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   944
    by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   945
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   946
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   947
lemma interior_complement: "interior(UNIV - S) = UNIV - closure(S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   948
  unfolding closure_interior
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   949
  by blast
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   950
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   951
subsection{* Frontier (aka boundary) *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   952
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   953
definition "frontier S = closure S - interior S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   954
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   955
lemma frontier_closed: "closed(frontier S)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   956
  by (simp add: frontier_def closed_diff closed_closure)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   957
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   958
lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(UNIV - S))"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   959
  by (auto simp add: frontier_def interior_closure)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   960
31420
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   961
lemma frontier_straddle:
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   962
  fixes a :: "'a::metric_space"
4c22ef11078b generalize type of islimpt
huffman
parents: 31418
diff changeset
   963
  shows "a \<in> frontier S \<longleftrightarrow> (\<forall>e>0. (\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e))" (is "?lhs \<longleftrightarrow> ?rhs")
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   964
proof
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   965
  assume "?lhs"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   966
  { fix e::real
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   967
    assume "e > 0"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   968
    let ?rhse = "(\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   969
    { assume "a\<in>S"
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   970
      have "\<exists>x\<in>S. dist a x < e" using `e>0` `a\<in>S` by(rule_tac x=a in bexI) auto
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   971
      moreover have "\<exists>x. x \<notin> S \<and> dist a x < e" using `?lhs` `a\<in>S`
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   972
	unfolding frontier_closures closure_def islimpt_def using `e>0`
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   973
	by (auto, erule_tac x="ball a e" in allE, auto)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   974
      ultimately have ?rhse by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   975
    }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   976
    moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   977
    { assume "a\<notin>S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   978
      hence ?rhse using `?lhs`
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   979
	unfolding frontier_closures closure_def islimpt_def
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   980
	using open_ball[of a e] `e > 0`
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
   981
	by (auto, erule_tac x = "ball a e" in allE, auto) (* FIXME: VERY slow! *)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   982
    }
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
   983
    ultimately have ?rhse by auto
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   984
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   985
  thus ?rhs by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   986
next
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   987
  assume ?rhs
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   988
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   989
  { fix T assume "a\<notin>S" and
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   990
    as:"\<forall>e>0. (\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e)" "a \<notin> S" "a \<in> T" "open T"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   991
    from `open T` `a \<in> T` have "\<exists>e>0. ball a e \<subseteq> T" unfolding open_contains_ball[of T] by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   992
    then obtain e where "e>0" "ball a e \<subseteq> T" by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   993
    then obtain y where y:"y\<in>S" "dist a y < e"  using as(1) by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   994
    have "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> a"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   995
      using `dist a y < e` `ball a e \<subseteq> T` unfolding ball_def using `y\<in>S` `a\<notin>S` by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   996
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   997
  hence "a \<in> closure S" unfolding closure_def islimpt_def using `?rhs` by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   998
  moreover
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
   999
  { fix T assume "a \<in> T"  "open T" "a\<in>S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1000
    then obtain e where "e>0" and balle: "ball a e \<subseteq> T" unfolding open_contains_ball using `?rhs` by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1001
    obtain x where "x \<notin> S" "dist a x < e" using `?rhs` using `e>0` by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1002
    hence "\<exists>y\<in>UNIV - S. y \<in> T \<and> y \<noteq> a" using balle `a\<in>S` unfolding ball_def by (rule_tac x=x in bexI)auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1003
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1004
  hence "a islimpt (UNIV - S) \<or> a\<notin>S" unfolding islimpt_def by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1005
  ultimately show ?lhs unfolding frontier_closures using closure_def[of "UNIV - S"] by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1006
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1007
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
  1008
lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1009
  by (metis frontier_def closure_closed Diff_subset)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1010
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1011
lemma frontier_empty: "frontier {} = {}"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1012
  by (simp add: frontier_def closure_empty)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1013
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1014
lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1015
proof-
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1016
  { assume "frontier S \<subseteq> S"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1017
    hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1018
    hence "closed S" using closure_subset_eq by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1019
  }
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1020
  thus ?thesis using frontier_subset_closed[of S] by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1021
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1022
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
  1023
lemma frontier_complement: "frontier(UNIV - S) = frontier S"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1024
  by (auto simp add: frontier_def closure_complement interior_complement)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1025
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1026
lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
  1027
  using frontier_complement frontier_subset_eq[of "UNIV - S"]
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1028
  unfolding open_closed by auto
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1029
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1030
subsection{* Common nets and The "within" modifier for nets. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1031
31285
0a3f9ee4117c generalize dist function to class real_normed_vector
huffman
parents: 31275
diff changeset
  1032
definition
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1033
  at_infinity :: "'a::real_normed_vector net" where
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1034
  "at_infinity = Abs_net (range (\<lambda>r. {x. r \<le> norm x}))"
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1035
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1036
definition
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1037
  indirection :: "real ^'n::finite \<Rightarrow> real ^'n \<Rightarrow> (real ^'n) net" (infixr "indirection" 70) where
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1038
  "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = c*s v}"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1039
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1040
text{* Prove That They are all nets. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1041
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1042
lemma Rep_net_at_infinity:
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1043
  "Rep_net at_infinity = range (\<lambda>r. {x. r \<le> norm x})"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1044
unfolding at_infinity_def
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1045
apply (rule Abs_net_inverse')
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1046
apply (rule image_nonempty, simp)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1047
apply (clarsimp, rename_tac r s)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1048
apply (rule_tac x="max r s" in exI, auto)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1049
done
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1050
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1051
lemma within_UNIV: "net within UNIV = net"
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1052
  by (simp add: Rep_net_inject [symmetric] Rep_net_within)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1053
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1054
subsection{* Identify Trivial limits, where we can't approach arbitrarily closely. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1055
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1056
definition
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1057
  trivial_limit :: "'a net \<Rightarrow> bool" where
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1058
  "trivial_limit net \<longleftrightarrow> {} \<in> Rep_net net"
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1059
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1060
lemma trivial_limit_within:
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1061
  shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1062
proof
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1063
  assume "trivial_limit (at a within S)"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1064
  thus "\<not> a islimpt S"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1065
    unfolding trivial_limit_def
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1066
    unfolding Rep_net_within Rep_net_at
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1067
    unfolding islimpt_def open_def [symmetric]
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1068
    apply (clarsimp simp add: expand_set_eq)
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1069
    apply (rename_tac T, rule_tac x=T in exI)
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1070
    apply (clarsimp, drule_tac x=y in spec, simp)
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1071
    done
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1072
next
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1073
  assume "\<not> a islimpt S"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1074
  thus "trivial_limit (at a within S)"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1075
    unfolding trivial_limit_def
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1076
    unfolding Rep_net_within Rep_net_at
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1077
    unfolding islimpt_def open_def [symmetric]
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1078
    apply (clarsimp simp add: image_image)
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1079
    apply (rule_tac x=T in image_eqI)
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1080
    apply (auto simp add: expand_set_eq)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1081
    done
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1082
qed
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1083
31391
97a2a3d4088e generalize type of 'at' to metric_space
huffman
parents: 31390
diff changeset
  1084
lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1085
  using trivial_limit_within [of a UNIV]
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1086
  by (simp add: within_UNIV)
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1087
31391
97a2a3d4088e generalize type of 'at' to metric_space
huffman
parents: 31390
diff changeset
  1088
lemma trivial_limit_at:
97a2a3d4088e generalize type of 'at' to metric_space
huffman
parents: 31390
diff changeset
  1089
  fixes a :: "'a::perfect_space"
97a2a3d4088e generalize type of 'at' to metric_space
huffman
parents: 31390
diff changeset
  1090
  shows "\<not> trivial_limit (at a)"
97a2a3d4088e generalize type of 'at' to metric_space
huffman
parents: 31390
diff changeset
  1091
  by (simp add: trivial_limit_at_iff)
97a2a3d4088e generalize type of 'at' to metric_space
huffman
parents: 31390
diff changeset
  1092
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1093
lemma trivial_limit_at_infinity:
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1094
  "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,zero_neq_one}) net)"
31391
97a2a3d4088e generalize type of 'at' to metric_space
huffman
parents: 31390
diff changeset
  1095
  (* FIXME: find a more appropriate type class *)
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1096
  unfolding trivial_limit_def Rep_net_at_infinity
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1097
  apply (clarsimp simp add: expand_set_eq)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1098
  apply (drule_tac x="scaleR r (sgn 1)" in spec)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1099
  apply (simp add: norm_scaleR norm_sgn)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1100
  done
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1101
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1102
lemma trivial_limit_sequentially: "\<not> trivial_limit sequentially"
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1103
  by (auto simp add: trivial_limit_def Rep_net_sequentially)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1104
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1105
subsection{* Some property holds "sufficiently close" to the limit point. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1106
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1107
lemma eventually_at: (* FIXME: this replaces Limits.eventually_at *)
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1108
  "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1109
unfolding eventually_at dist_nz by auto
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1110
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1111
lemma eventually_at_infinity:
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1112
  "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1113
unfolding eventually_def Rep_net_at_infinity by auto
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1114
31346
fa93996e9572 generalize at function to class perfect_space
huffman
parents: 31345
diff changeset
  1115
lemma eventually_within: "eventually P (at a within S) \<longleftrightarrow>
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1116
        (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
31393
b8570dead501 reuse definition of nets from Limits.thy
huffman
parents: 31391
diff changeset
  1117
unfolding eventually_within eventually_at dist_nz by auto
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1118
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1119
lemma eventually_within_le: "eventually P (at a within S) \<longleftrightarrow>
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1120
        (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> P x)" (is "?lhs = ?rhs")
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1121
unfolding eventually_within
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1122
apply safe
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1123
apply (rule_tac x="d/2" in exI, simp)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1124
apply (rule_tac x="d" in exI, simp)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1125
done
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1126
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1127
lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1128
  unfolding eventually_def trivial_limit_def
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1129
  using Rep_net_nonempty [of net] by auto
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1130
31347
357d58c5743a new lemmas about eventually; rewrite Lim proofs to use more abstract properties of eventually
huffman
parents: 31346
diff changeset
  1131
lemma always_eventually: "(\<forall>x. P x) ==> eventually P net"
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1132
  unfolding eventually_def trivial_limit_def
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1133
  using Rep_net_nonempty [of net] by auto
31347
357d58c5743a new lemmas about eventually; rewrite Lim proofs to use more abstract properties of eventually
huffman
parents: 31346
diff changeset
  1134
31348
738eb25e1dd8 more abstract properties of eventually
huffman
parents: 31347
diff changeset
  1135
lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1136
  unfolding trivial_limit_def eventually_def by auto
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1137
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1138
lemma eventually_False: "eventually (\<lambda>x. False) net \<longleftrightarrow> trivial_limit net"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1139
  unfolding trivial_limit_def eventually_def by auto
31348
738eb25e1dd8 more abstract properties of eventually
huffman
parents: 31347
diff changeset
  1140
738eb25e1dd8 more abstract properties of eventually
huffman
parents: 31347
diff changeset
  1141
lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1142
  apply (safe elim!: trivial_limit_eventually)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1143
  apply (simp add: eventually_False [symmetric])
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1144
  done
31348
738eb25e1dd8 more abstract properties of eventually
huffman
parents: 31347
diff changeset
  1145
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1146
text{* Combining theorems for "eventually" *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1147
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1148
lemma eventually_conjI:
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1149
  "\<lbrakk>eventually (\<lambda>x. P x) net; eventually (\<lambda>x. Q x) net\<rbrakk>
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1150
    \<Longrightarrow> eventually (\<lambda>x. P x \<and> Q x) net"
31393
b8570dead501 reuse definition of nets from Limits.thy
huffman
parents: 31391
diff changeset
  1151
by (rule eventually_conj)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1152
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1153
lemma eventually_rev_mono:
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1154
  "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1155
using eventually_mono [of P Q] by fast
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1156
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1157
lemma eventually_and: " eventually (\<lambda>x. P x \<and> Q x) net \<longleftrightarrow> eventually P net \<and> eventually Q net"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1158
  by (auto intro!: eventually_conjI elim: eventually_rev_mono)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1159
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1160
lemma eventually_false: "eventually (\<lambda>x. False) net \<longleftrightarrow> trivial_limit net"
31390
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1161
  by (auto simp add: eventually_False)
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1162
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1163
lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"
1d0478b16613 redefine nets as filter bases
huffman
parents: 31348
diff changeset
  1164
  by (simp add: eventually_False)
31347
357d58c5743a new lemmas about eventually; rewrite Lim proofs to use more abstract properties of eventually
huffman
parents: 31346
diff changeset
  1165
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1166
subsection{* Limits, defined as vacuously true when the limit is trivial. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1167
31393
b8570dead501 reuse definition of nets from Limits.thy
huffman
parents: 31391
diff changeset
  1168
notation tendsto (infixr "--->" 55)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1169
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1170
  text{* Notation Lim to avoid collition with lim defined in analysis *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1171
definition "Lim net f = (THE l. (f ---> l) net)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1172
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
  1173
lemma Lim:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1174
 "(f ---> l) net \<longleftrightarrow>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1175
        trivial_limit net \<or>
31348
738eb25e1dd8 more abstract properties of eventually
huffman
parents: 31347
diff changeset
  1176
        (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
738eb25e1dd8 more abstract properties of eventually
huffman
parents: 31347
diff changeset
  1177
  unfolding tendsto_def trivial_limit_eq by auto
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1178
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1179
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1180
text{* Show that they yield usual definitions in the various cases. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1181
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1182
lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1183
           (\<forall>e>0. \<exists>d>0. \<forall>x\<in>S. 0 < dist x a  \<and> dist x a  <= d \<longrightarrow> dist (f x) l < e)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1184
  by (auto simp add: tendsto_def eventually_within_le)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1185
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1186
lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1187
        (\<forall>e >0. \<exists>d>0. \<forall>x \<in> S. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1188
  by (auto simp add: tendsto_def eventually_within)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1189
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1190
lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1191
        (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1192
  by (auto simp add: tendsto_def eventually_at)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1193
31342
b7941738e3a1 add lemmas Lim_at_iff_LIM, Lim_sequentially_iff_LIMSEQ
huffman
parents: 31341
diff changeset
  1194
lemma Lim_at_iff_LIM: "(f ---> l) (at a) \<longleftrightarrow> f -- a --> l"
b7941738e3a1 add lemmas Lim_at_iff_LIM, Lim_sequentially_iff_LIMSEQ
huffman
parents: 31341
diff changeset
  1195
  unfolding Lim_at LIM_def by (simp only: zero_less_dist_iff)
b7941738e3a1 add lemmas Lim_at_iff_LIM, Lim_sequentially_iff_LIMSEQ
huffman
parents: 31341
diff changeset
  1196
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1197
lemma Lim_at_infinity:
30582
638b088bb840 imported patch euclidean
huffman
parents: 30549
diff changeset
  1198
  "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x::real^'n::finite. norm x >= b \<longrightarrow> dist (f x) l < e)"
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1199
  by (auto simp add: tendsto_def eventually_at_infinity)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1200
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30268
diff changeset
  1201
lemma Lim_sequentially:
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1202
 "(S ---> l) sequentially \<longleftrightarrow>
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1203
          (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (S n) l < e)"
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1204
  by (auto simp add: tendsto_def eventually_sequentially)
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1205
31342
b7941738e3a1 add lemmas Lim_at_iff_LIM, Lim_sequentially_iff_LIMSEQ
huffman
parents: 31341
diff changeset
  1206
lemma Lim_sequentially_iff_LIMSEQ: "(S ---> l) sequentially \<longleftrightarrow> S ----> l"
b7941738e3a1 add lemmas Lim_at_iff_LIM, Lim_sequentially_iff_LIMSEQ
huffman
parents: 31341
diff changeset
  1207
  unfolding Lim_sequentially LIMSEQ_def ..
b7941738e3a1 add lemmas Lim_at_iff_LIM, Lim_sequentially_iff_LIMSEQ
huffman
parents: 31341
diff changeset
  1208
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1209
lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"
31348
738eb25e1dd8 more abstract properties of eventually
huffman
parents: 31347
diff changeset
  1210
  unfolding tendsto_def by (auto elim: eventually_rev_mono)
30262
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1211
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1212
text{* The expected monotonicity property. *}
5794fee816c3 A formalization of Topology on Euclidean spaces, Includes limits (nets) , continuity, fixpoint theorems, homeomorphisms
chaieb
parents:
diff changeset
  1213
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1214
lemma Lim_within_empty: "(f ---> l) (net within {})"
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1215
  unfolding tendsto_def Limits.eventually_within by simp
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1216
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1217
lemma Lim_within_subset: "(f ---> l) (net within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (net within T)"
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1218
  unfolding tendsto_def Limits.eventually_within
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1219
  by (auto elim!: eventually_elim1)
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1220
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31445
diff changeset
  1221
lemma Lim_Un: assumes "(f ---> l) (net within S)" "(f ---> l) (net within T)"